Dual shape functions

For all considered coupled multiphysics problems which are going to be involved in the nodal information transfer within this thesis, the same solution space S for the primary quantities of interest is employed. However, it is important to point out that there exist problems where this restriction is not valid due to stability reasons, like for example in fluid simulations. The solution space corresponds to an arbitrary continuous fieldsand reads

S ={s∈H¹(Ω)|s(X, t) = ˆs(X, t)on∂Ω_d}. (6.14) Herein,H¹(Ω)denotes again the well-known Sobolev space of functions with square integrable values and first derivatives. Moreover,Ωis the continuous volume on which the problem is de-fined and∂Ωdis its Dirichlet boundary with prescribed valuesˆs(X, t). For discretization in space into the considered non-matching meshesΩ_i fori = 1,2, standard isoparametric, Langrangian finite elements are employed. This defines the usual finite dimensional subsetsS^i,h fori= 1,2, that are approximations ofS.

As it is aimed at solving monolithic systems of equations for coupled multiphysics problems, a globally assembled projection operator is required. At this point, it becomes obvious that in-verting the first mortar matricesD_ifori= 1,2in Equations (6.6) and (6.7) constitutes the major drawback of the standard mortar approach. For the mortar projection operators, any single nodal value on one mesh globally affects the entire other mesh and vice versa. However, there is a pos-sibility to localize the influence domain (support) onto which a certain nodal value is mapped.

Algebraically, this localized form is characterized by the mortar matricesD_ifori= 1,2 becom-ing diagonal. This can be achieved by employbecom-ing the already introduced dual shape functions which are based on a biorthogonal relationship with the standard shape functions for the in-terpolation of the dual fields d_i for i = 1,2, see Section 3.4.1.2. Within mortar methods for computational contact problems and wear modeling these dual shape functions have been suc-cessfully employed in Chapters 3, 4 and 5. However, in the following, these dual shape functions will be extended towards 3D problems.

6.2.3.1. Extension to 3D problems

For 4-node tetrahedral elements in 3D, dual shape functions have been used in Lamichhane [148], Lamichhane et al. [149] and Tkachuk and Bischoff [272]. However, up to the best knowl-edge of the author, dual shape functions for all other types of 3D Lagrangian finite elements have not yet been analyzed in recent publications. In order to be able to vary not only the mesh ratio, but also the element types and their polynomial order, dual shape functions for commonly used first- and second-order finite elements need to be considered in the following. The basic construction methodologies as described in Section 3.4.1.2 are still valid but will be shortly re-viewed here. Thus, the biorthogonality condition for dual shape functions can be realized as linear combination of standard shape functions. Again, this can be written as multiplication of a vector containing standard shape functions within each element with an element coefficient matrixC_e:

Φj(ξ) = cjkNk(ξ), C_e = [cjk]∈R^ne×ne . (6.15) The coefficient matrix itself reads

C_e =D_eM⁻_e¹, (6.16)

D_e = [djk]∈R^ne×ne, djk =δjk

Z

eNk(ξ)J(ξ)de, (6.17) M_e = [mjk]∈R^ne×ne, mjk =

Z

eNj(ξ)Nk(ξ)J(ξ)de. (6.18) All these steps are based on the assumption that the number of nodes carrying discrete unknowns from the abstract dual fieldsdiare equal to the nodes of the fieldssi, fori= 1,2. The only differ-ence compared to the construction rules for 2D elements is the integration of a volume element instead of a surface element. For the sake of clearness, the dual shape function of one node is shown for a regular hex8 element in Figure 6.3. Here, a dual shape function and a standard shape function are visualized for the red marked node at the bottom left with help of 4 characteristic slides (a-d) through the element. It can be nicely seen that the green shape function is of zero value at all nodes except its own (red) node, where a positive value occurs (not visualized in

6.2. Fundamentals on volume projection of nodal information

Figure 6.3: Standard (green) and dual (red) shape functions of the red node for an undistorted 4-node hexahedral element. Visualization is realized at 4 slices a-d.

Figure 6.3). When going from slide a to d, the value at the red node continuously decreases till it is zero. This corresponds to the general knowledge of standard shape functions. The dual shape function behaves completely different. It is discontinuous at the element boundaries and have higher values at the nodes. In addition, it changes the sign of its values, which is also well-known for 2D dual shape functions, see Popp et al. [213]. Special treatment of second-order shape functions in 3D is discussed in the following subsection.

6.2.3.2. Basis transformation for second-order elements

The creation of dual shape functions relies on the fact that the integrals of standard shape func-tions in (6.17) and (6.18) lead to non-zero values. Whereas this condition is readily fulfilled for first- and second-order elements for the integral in (6.18), non-zero values for the integral in (6.17) are only assured for first-order interpolation. This becomes obvious for a second-order, i.e. 10-node, tetrahedral element that is slightly distorted as shown in Figure 6.4. Here, the three edge nodes are relocated in such a way that the integral of the shape function associated with the top corner node vanishes. To overcome this problem, a simple basis transformation is employed for the standard shape functions as previously introduced for surface elements in the context of mortar contact algorithms, see Popp et al. [213]. The basic idea is to shift shape function values from edge to corner nodes to guarantee not only non-zero integral values in (6.17), but integral positivity. This is possible because the shape functions associated with the edge nodes are strictly positive for 10-node tetrahedral (tet10) elements. The modified shape functions due to the basis transformation are denoted withN˜. The corresponding dual shape functions are then built based

undeformed state deformed state

Figure 6.4: Critical scenario for a second-order (10-node) tetrahedral element: the undeformed domain represents the reference tetrahedron. In the deformed state, the marked edge nodes are moved downwards. The deformed state leads to a zero value for the stan-dard shape function of the top node. Figure taken from Farah et al. [70].

on the transformed shape functions, viz.

Z There are a lot of possible basis transformations available, which can all sufficiently shift shape function values in a way that the integral positivity condition is fulfilled. In the following, one simple basis transformation is exemplarily shown for a tet10 element. This simple basis trans-formation will be employed for all numerical examples with second-order interpolation in this chapter. Based on a node numbering, where first all corner nodes are listed and afterwards all edge nodes, the basis transformation for a tet10 element reads

hN˜¹ N˜² N˜³ N˜⁴ N˜⁵ N˜⁶ N˜⁷ N˜⁸ N˜⁹ N˜¹⁰ⁱ=

6.2. Fundamentals on volume projection of nodal information This transformation matrix is symmetric, since it shifts edge node contributions equally to the ad-jacent corner nodes. Furthermore, partition of unity is assured, which simplifies the construction of the corresponding dual shape functions. The transformation parameter αq has to be chosen large enough to guarantee integral positivity of the edge node shape functions, but is obviously limited byαq < ¹₂. Based on these considerations and the numerical experience, the transforma-tion parameter is defined asαq = ¹₃. By modifying the standard shape functions, the interpola-tion of the continuous fieldss_ifori= 1,2can be alternatively expressed with basis transformed nodal quantities

si =N_i s_i =N˜_i˜s_i. (6.21)

A consistent node-wise assembly of the element transformation matrix T_e in (6.20) yields the global transformation matricesT_i for quantities on meshesΩifori= 1,2, viz.

s_i =T_i˜s_i. (6.22)

The global transformation matrices are sparse and have the same dimensions as the correspond-ingDmatrices. Applying the basis transformation for the duality pairing in (6.3) yields

hdi, sii^Ωi =d^T_i D˜_i˜s_i =d^T_i D˜_i T⁻¹_i s_i. (6.23) Here, the mortar matrices D˜_i are square and diagonal due to the biorthogonality condition in (6.19) for the modified shape functions. Now, the final mortar projection operators for the non-modified quantities can be simply formulated by multiplying the transformation matrix and the basis-transformed projection operator:

s₂ =T₂D˜⁻₂¹M₂₁s₁ =T₂ P˜^m₂₁s₁ =P^m₂₁s₁, (6.24) s₁ =T₁D˜⁻₁¹M₁₂s₂ =T₁ P˜^m₁₂s₂ =P^m₁₂s₂. (6.25) With this modification at hand, the mortar approach for information transfer can be robustly employed for general second-order finite elements without loss of computational efficiency.